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 January 27th, 2012, 03:28 PM #1 Member   Joined: Jan 2012 Posts: 63 Thanks: 0 continuous Suppose f:R->R and g:R->R are continuous functions, and define a new function h:R->R by h(x)=max{f(x),g(x)}. Is h continuous? Why? I don't know how to begin; I think I should discuss different situation because f can be greater then g or smaller than g or they have intersection points. January 27th, 2012, 10:34 PM #2 Math Team   Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: continuous Well, consider the graphs of f and g on the same page. If one function is above the other, say f above g, then h = f and is continuous because f is continuous. If for some x values g is above, for other x values f is above, they must intersect. The intersection will be the max and belong to h. The other parts of h will be from either f or g depending which is on top or other intersections. Ether way h is continuous, though may be composed of different parts of f and g. The point here is that if 2 continuous functions cross each other they must intersect in at least 1 point. Another good question to ask about given the stated conditions...is h(x) differentiable everywhere? Why? January 31st, 2012, 02:19 AM   #3
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Re: continuous

Quote:
 Originally Posted by frankpupu Suppose f:R->R and g:R->R are continuous functions, and define a new function h:R->R by h(x)=max{f(x),g(x)}. Is h continuous? Why? I don't know how to begin; I think I should discuss different situation because f can be greater then g or smaller than g or they have intersection points.

Take : f(x) = 0 ,which is a continuous function ,and g(x) = sinx , or g(x) = cos x,then the max{f(x),g(x)} are just point above the x- axis.

Can this function be continuous ?? January 31st, 2012, 04:27 AM #4 Math Team   Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 Re: continuous They are both continuous everywhere but not differentiable everywhere. You lose the derivative whenever they intersect. if f(x)=0 then parts of the x axis belong to h(x), precisely those parts where g(x) <= 0 http://www.wolframalpha.com/input/?i=ma ... ual=Submit http://www.wolframalpha.com/input/?i=ma ... ual=Submit January 31st, 2012, 05:17 AM   #5
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Re: continuous

Quote:
 Originally Posted by outsos Can this function be continuous ??
Yes, not all points are above x-axis.
Quote:
 Originally Posted by outsos Take f(x) = 0 ,which is a continuous function ,and g(x) = sinx
Gives h(x) = max{f(x),g(x)} = max{0, sin(x)}
Now, h(0) = max{0,sin(0)} is above the x-axis? Tags continuous Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post frankpupu Calculus 2 February 9th, 2012 05:43 PM thedoctor818 Real Analysis 17 November 9th, 2010 08:19 AM guroten Real Analysis 2 November 5th, 2010 01:46 PM summerset353 Real Analysis 2 February 22nd, 2010 03:37 PM babyRudin Real Analysis 6 October 24th, 2008 12:58 AM

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